Mathematics

15 pages, 29218 KiB

Open AccessEditor’s ChoiceArticle

Coexisting Attractor in a Gyrostat Chaotic System via Basin of Attraction and Synchronization of Two Nonidentical Mechanical Systems

by Muhammad Marwan, Vagner Dos Santos, Muhammad Zainul Abidin and Anda Xiong

Mathematics 2022, 10(11), 1914; https://doi.org/10.3390/math10111914 - 2 Jun 2022

Cited by 11 | Viewed by 3037

Abstract

This paper is divided into two main portions. First, we look at basins of attraction as a tool with a unique set of characteristics for discussing multistability and coexisting attractors in a gyrostat chaotic system. For the validation of coexisting attractors in different [...] Read more.

This paper is divided into two main portions. First, we look at basins of attraction as a tool with a unique set of characteristics for discussing multistability and coexisting attractors in a gyrostat chaotic system. For the validation of coexisting attractors in different basins, several approaches such as bifurcation diagrams, Lyapunov exponents, and the Poincaré section are applied. The second half of the study synchronizes two mechanical chaotic systems using a novel controller, with gyrostat and quadrotor unmanned aerial vehicle (QUAV) chaotic systems acting as master and slave systems, respectively. The error dynamical system and the parameter updated law are built using Lyapunov’s theory, and it is discovered that under certain parametric conditions, the trajectories of the QUAV chaotic system overlap and begin to match the features of the gyrostat chaotic system. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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74 pages, 751 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Centrally Essential Rings and Semirings

by Askar Tuganbaev

Mathematics 2022, 10(11), 1867; https://doi.org/10.3390/math10111867 - 30 May 2022

Viewed by 2206

Abstract

This paper is a survey of results on centrally essential rings and semirings. A ring (respectively, semiring) is said to be centrally essential if it is either commutative or satisfies the property that for any non-central element a, there exist non-zero central [...] Read more.

This paper is a survey of results on centrally essential rings and semirings. A ring (respectively, semiring) is said to be centrally essential if it is either commutative or satisfies the property that for any non-central element a, there exist non-zero central elements x and y with ax = y. The class of centrally essential rings is very large; many corresponding examples are given in the work. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

15 pages, 1612 KiB

Open AccessEditor’s ChoiceArticle

A Preventive Replacement Policy for a System Subject to Bivariate Generalized Polya Failure Process

by Hyunju Lee, Ji Hwan Cha and Maxim Finkelstein

Mathematics 2022, 10(11), 1833; https://doi.org/10.3390/math10111833 - 26 May 2022

Cited by 2 | Viewed by 1863

Abstract

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover [...] Read more.

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover this important in-practice setting. Therefore, our paper is the first to deal with these issues by modeling dependence in the bivariate set up when a system consists of two dependent parts. We employ the bivariate generalized Polya process to model the corresponding failure and repair process. Relevant stochastic properties of this process have been obtained in order to propose and further discuss the new optimal bivariate preventive maintenance policy with two decision parameters: age and operational history. Moreover, introducing these two parameters in the considered context is also a new feature of the study. Under the proposed policy, the long-run average cost rate is derived and the optimal replacement policies are investigated. Detailed numerical examples illustrate our findings and show the potential efficiency of the obtained results in practice. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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14 pages, 3760 KiB

Open AccessEditor’s ChoiceArticle

An Efficient Computational Technique for the Electromagnetic Scattering by Prolate Spheroids

by Ludovica Tognolatti, Cristina Ponti, Massimo Santarsiero and Giuseppe Schettini

Mathematics 2022, 10(10), 1761; https://doi.org/10.3390/math10101761 - 21 May 2022

Cited by 3 | Viewed by 2622

Abstract

In this paper we present an efficient Matlab computation of a 3-D electromagnetic scattering problem, in which a plane wave impinges with a generic inclination onto a conducting ellipsoid of revolution. This solid is obtained by the rotation of an ellipse around one [...] Read more.

In this paper we present an efficient Matlab computation of a 3-D electromagnetic scattering problem, in which a plane wave impinges with a generic inclination onto a conducting ellipsoid of revolution. This solid is obtained by the rotation of an ellipse around one of its axes, which is also known as a spheroid. We have developed a fast and ad hoc code to solve the electromagnetic scattering problem, using spheroidal vector wave functions, which are special functions used to describe physical problems in which a prolate or oblate spheroidal reference system is considered. Numerical results are presented, both for TE and TM polarization of the incident wave, and are validated by a comparison with results obtained by a commercial electromagnetic simulator. Full article

(This article belongs to the Special Issue Analytical Methods in Wave Scattering and Diffraction)

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14 pages, 1158 KiB

Open AccessEditor’s ChoiceArticle

State Estimation for Complex-Valued Inertial Neural Networks with Multiple Time Delays

by Yaning Yu and Ziye Zhang

Mathematics 2022, 10(10), 1725; https://doi.org/10.3390/math10101725 - 18 May 2022

Cited by 10 | Viewed by 2299

Abstract

In this paper, the problem of state estimation for complex-valued inertial neural networks with leakage, additive and distributed delays is considered. By means of the Lyapunov–Krasovskii functional method, the Jensen inequality, and the reciprocally convex approach, a delay-dependent criterion based on linear matrix [...] Read more.

In this paper, the problem of state estimation for complex-valued inertial neural networks with leakage, additive and distributed delays is considered. By means of the Lyapunov–Krasovskii functional method, the Jensen inequality, and the reciprocally convex approach, a delay-dependent criterion based on linear matrix inequalities (LMIs) is derived. At the same time, the network state is estimated by observing the output measurements to ensure the global asymptotic stability of the error system. Finally, two examples are given to verify the effectiveness of the proposed method. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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25 pages, 414 KiB

Open AccessEditor’s ChoiceArticle

Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations

by Shaomin Li, Haoyu Wei and Xiaoyu Lei

Mathematics 2022, 10(10), 1700; https://doi.org/10.3390/math10101700 - 16 May 2022

Cited by 3 | Viewed by 2633

Abstract

Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion [...] Read more.

Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion estimation for the heterogeneous NBR models, which model the dispersion parameter as a function. Specifically, we proposed a double regression and applied a double

ℓ_{1}

-penalty to both regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for the lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demonstrate that the new methods are effective. Full article

(This article belongs to the Special Issue New Advances in High-Dimensional and Non-asymptotic Statistics)

22 pages, 1091 KiB

Open AccessEditor’s ChoiceArticle

Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures

by Klaus Gürlebeck, Dmitrii Legatiuk and Kemmar Webber

Mathematics 2022, 10(10), 1670; https://doi.org/10.3390/math10101670 - 13 May 2022

Cited by 1 | Viewed by 1999

Abstract

The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and [...] Read more.

The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)

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16 pages, 359 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Fisher, Bayes, and Predictive Inference

by Sandy Zabell

Mathematics 2022, 10(10), 1634; https://doi.org/10.3390/math10101634 - 11 May 2022

Cited by 2 | Viewed by 2870

Abstract

We review historically the position of Sir R.A. Fisher towards Bayesian inference and, particularly, the classical Bayes–Laplace paradigm. We focus on his Fiducial Argument. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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10 pages, 431 KiB

Open AccessEditor’s ChoiceArticle

Andness Directedness for t-Norms and t-Conorms

by Vicenç Torra

Mathematics 2022, 10(9), 1598; https://doi.org/10.3390/math10091598 - 8 May 2022

Cited by 2 | Viewed by 2404

Abstract

Tools for decision making need to be simple to use. In previous papers, we advocated that decision engineering needs to provide these tools, as well as a list of necessary properties that aggregation functions need to satisfy. When we model decisions using aggregation [...] Read more.

Tools for decision making need to be simple to use. In previous papers, we advocated that decision engineering needs to provide these tools, as well as a list of necessary properties that aggregation functions need to satisfy. When we model decisions using aggregation functions, andness-directedness is one of them. A crucial aspect in any decision is the degree of compromise between criteria. Given an aggregation function, andness establishes to what degree the function behaves in a conjunctive manner. That is, to what degree some criteria are mandatory. Nevertheless, from an engineering perspective, what we know is that some criteria are strongly required and we cannot ignore a bad evaluation even when other criteria are correctly evaluated. That is, given our requirements of andness, what are the aggregation functions we need to select. Andness is not only for mean-like functions, but it also applies to t-norms and t-conorms. In this paper, we study this problem and show how to select t-norms and t-conorms based on the andness level. Full article

(This article belongs to the Special Issue Fuzzy Sets and Artificial Intelligence)

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24 pages, 3902 KiB

Open AccessEditor’s ChoiceArticle

A Safe and Efficient Lane Change Decision-Making Strategy of Autonomous Driving Based on Deep Reinforcement Learning

by Kexuan Lv, Xiaofei Pei, Ci Chen and Jie Xu

Mathematics 2022, 10(9), 1551; https://doi.org/10.3390/math10091551 - 5 May 2022

Cited by 30 | Viewed by 5363

Abstract

As an indispensable branch of machine learning (ML), reinforcement learning (RL) plays a prominent role in the decision-making process of autonomous driving (AD), which enables autonomous vehicles (AVs) to learn an optimal driving strategy through continuous interaction with the environment. This paper proposes [...] Read more.

As an indispensable branch of machine learning (ML), reinforcement learning (RL) plays a prominent role in the decision-making process of autonomous driving (AD), which enables autonomous vehicles (AVs) to learn an optimal driving strategy through continuous interaction with the environment. This paper proposes a deep reinforcement learning (DRL)-based motion planning strategy for AD tasks in the highway scenarios where an AV merges into two-lane road traffic flow and realizes the lane changing (LC) maneuvers. We integrate the DRL model into the AD system relying on the end-to-end learning method. An improved DRL algorithm based on deep deterministic policy gradient (DDPG) is developed with well-defined reward functions. In particular, safety rules (SR), safety prediction (SP) module and trauma memory (TM) as well as the dynamic potential-based reward shaping (DPBRS) function are adopted to further enhance safety and accelerate learning of the LC behavior. For validation, the proposed DSSTD algorithm is trained and tested on the dual-computer co-simulation platform. The comparative experimental results show that our proposal outperforms other benchmark algorithms in both driving safety and efficiency. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods, Machine Learning and Deep Learning Based Applications)

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16 pages, 697 KiB

Open AccessEditor’s ChoiceArticle

Fixed-Time Synchronization for Fuzzy-Based Impulsive Complex Networks

by Lu Pang, Cheng Hu, Juan Yu and Haijun Jiang

Mathematics 2022, 10(9), 1533; https://doi.org/10.3390/math10091533 - 3 May 2022

Cited by 9 | Viewed by 2154

Abstract

This paper mainly deals with the issue of fixed-time synchronization of fuzzy-based impulsive complex networks. By developing fixed-time stability of impulsive systems and proposing a T-S fuzzy control strategy with pure power-law form, some simple criteria are acquired to achieve fixed-time synchronization of [...] Read more.

This paper mainly deals with the issue of fixed-time synchronization of fuzzy-based impulsive complex networks. By developing fixed-time stability of impulsive systems and proposing a T-S fuzzy control strategy with pure power-law form, some simple criteria are acquired to achieve fixed-time synchronization of fuzzy-based impulsive complex networks and the estimation of the synchronized time is given. Ultimately, the presented control scheme and synchronization criteria are verified by numerical simulation. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

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16 pages, 310 KiB

Open AccessEditor’s ChoiceArticle

Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Mathematics 2022, 10(9), 1524; https://doi.org/10.3390/math10091524 - 2 May 2022

Cited by 7 | Viewed by 3039

Abstract

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. [...] Read more.

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics)

43 pages, 606 KiB

Open AccessEditor’s ChoiceArticle

General Non-Local Continuum Mechanics: Derivation of Balance Equations

by Vasily E. Tarasov

Mathematics 2022, 10(9), 1427; https://doi.org/10.3390/math10091427 - 23 Apr 2022

Cited by 26 | Viewed by 2527

Abstract

In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing [...] Read more.

In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing mechanics of media with general form of nonlocality in space and time. Balance equations for mass, momentum, and energy, which describe conservation laws for nonlocal continuum, are derived by using the fundamental theorems of the GFC. The general balance equation in the integral form are derived by using the second fundamental theorems of the GFC. The first fundamental theorems of GFC and the proposed fractional analogue of the Titchmarsh theorem are used to derive the differential form of general balance equations from the integral form of balance equations. Using the general fractional vector calculus, the equations of conservation of mass, momentum, and energy are also suggested for a wide class of regions and surfaces. Full article

(This article belongs to the Section E4: Mathematical Physics)

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10 pages, 1798 KiB

Open AccessEditor’s ChoiceArticle

Confidence-Interval-Based Fuzzy Testing for the Lifetime Performance Index of Electronic Product

by Chun-Min Yu, Kuen-Suan Chen and Ting-Hsin Hsu

Mathematics 2022, 10(9), 1405; https://doi.org/10.3390/math10091405 - 22 Apr 2022

Cited by 6 | Viewed by 1749

Abstract

When the lifetime of an electronic component does not reach the required level, it can be enhanced by means of the paralleling current sharing backup system or the redundant backup system. The lifetime of the redundant backup system is the sum of lifetimes [...] Read more.

When the lifetime of an electronic component does not reach the required level, it can be enhanced by means of the paralleling current sharing backup system or the redundant backup system. The lifetime of the redundant backup system is the sum of lifetimes of all electronic components, which is the maximum of all the electronic components’ lifetimes, compared with the lifetime of the parallel current sharing backup system. For the purpose of enhancing products’ reliability, electronic goods are usually designed with spare electronic components. If it is assumed that there are

m - 1

redundant backup components for each electronic product, then the lifetime of the electronic product will be distributed as a Gamma distribution with two parameters—

m

and

λ

, where

λ

is the mean for each lifetime of each electronic component. According to numerous studies, the sample size is not large, as it takes a long time to test the lifetime of an electronic product, and enterprises consider cost and timeliness. This paper concerns the performance index of the lifetime of the electronic product. Therefore, based on the confidence interval, this paper aims to develop a fuzzy testing model. As this model can integrate past data and expert experience, the testing accuracy can be retained despite small-sized samples. In fact, through adopting the testing model proposed by this paper, companies can make precise and intelligent decisions instantly with the use of small-sized samples to grasp the opportunities for improvement. Full article

(This article belongs to the Special Issue Fuzzy Applications in Industrial Engineering)

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28 pages, 2417 KiB

Open AccessEditor’s ChoiceArticle

Using Markov-Switching Models in US Stocks Optimal Portfolio Selection in a Black–Litterman Context (Part 1)

by Oscar V. De la Torre-Torres, Evaristo Galeana-Figueroa, María de la Cruz Del Río-Rama and José Álvarez-García

Mathematics 2022, 10(8), 1296; https://doi.org/10.3390/math10081296 - 13 Apr 2022

Cited by 5 | Viewed by 2999

Abstract

In this study, we tested the benefit of using Markov-Switching (M-S) models to forecast the views of the 26 most traded stocks in the US in a Black–Litterman (B–L) optimal selection context. With weekly historical data of these stocks from 1 January 1980, [...] Read more.

In this study, we tested the benefit of using Markov-Switching (M-S) models to forecast the views of the 26 most traded stocks in the US in a Black–Litterman (B–L) optimal selection context. With weekly historical data of these stocks from 1 January 1980, we estimated and simulated (from 7 January 2000, to 7 February 2022) three portfolios that used M-S views in each stock and blended them with the market equilibrium views in a B–L context. Our position was that the B–L optimal portfolios could generate alpha (extra return) against a buy-and-hold and an actively managed portfolio with sample portfolio parameters (à la Markowitz, SampP). Our results suggest that the outperformance of the B–L managed portfolios holds only in the short term. In the long-term, the performance of the B–L portfolios, the SampP, and the market portfolio are statistically equal in terms of returns or their mean–variance efficiency in an ex-ante or ex-post analysis. Full article

(This article belongs to the Special Issue Markov-Chain Modelling and Applications)

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11 pages, 2485 KiB

Open AccessEditor’s ChoiceArticle

Edge of Chaos in Memristor Cellular Nonlinear Networks

by Angela Slavova and Ventsislav Ignatov

Mathematics 2022, 10(8), 1288; https://doi.org/10.3390/math10081288 - 12 Apr 2022

Cited by 3 | Viewed by 2395

Abstract

Information processing in the brain takes place in a dense network of neurons connected through synapses. The collaborative work between these two components (Synapses and Neurons) allows for basic brain functions such as learning and memorization. The so-called von Neumann bottleneck, which limits [...] Read more.

Information processing in the brain takes place in a dense network of neurons connected through synapses. The collaborative work between these two components (Synapses and Neurons) allows for basic brain functions such as learning and memorization. The so-called von Neumann bottleneck, which limits the information processing capability of conventional systems, can be overcome by the efficient emulation of these computational concepts. To this end, mimicking the neuronal architectures with silicon-based circuits, on which neuromorphic engineering is based, is accompanied by the development of new devices with neuromorphic functionalities. We shall study different memristor cellular nonlinear networks models. The rigorous mathematical analysis will be presented based on local activity theory, and the edge of chaos domain will be determined in the models under consideration. Simulations of these models working on the edge of chaos will show the generation of static and dynamic patterns. Full article

(This article belongs to the Special Issue Memristor Cellular Nonlinear Networks: Theory and Applications)

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16 pages, 304 KiB

Open AccessEditor’s ChoiceArticle

Limiting Distributions of a Non-Homogeneous Markov System in a Stochastic Environment in Continuous Time

by P. -C. G. Vassiliou

Mathematics 2022, 10(8), 1214; https://doi.org/10.3390/math10081214 - 7 Apr 2022

Cited by 2 | Viewed by 2502

Abstract

The stochastic process non-homogeneous Markov system in a stochastic environment in continuous time (S-NHMSC) is introduced in the present paper. The ordinary non-homogeneous Markov process is a very special case of an S-NHMSC. I studied the expected population structure of the S-NHMSC, the [...] Read more.

The stochastic process non-homogeneous Markov system in a stochastic environment in continuous time (S-NHMSC) is introduced in the present paper. The ordinary non-homogeneous Markov process is a very special case of an S-NHMSC. I studied the expected population structure of the S-NHMSC, the first central classical problem of finding the conditions under which the asymptotic behavior of the expected population structure exists and the second central problem of finding which expected relative population structures are possible limiting ones, provided that the limiting vector of input probabilities into the population is controlled. Finally, the rate of convergence was studied. Full article

(This article belongs to the Special Issue Probability and Stochastic Processes with Applications to Communications, Systems and Networks)

28 pages, 396 KiB

Open AccessEditor’s ChoiceArticle

On Numerical Approximations of the Koopman Operator

by Igor Mezić

Mathematics 2022, 10(7), 1180; https://doi.org/10.3390/math10071180 - 5 Apr 2022

Cited by 32 | Viewed by 4935

Abstract

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional [...] Read more.

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed. Full article

(This article belongs to the Special Issue Dynamical Systems and Operator Theory)

25 pages, 402 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Bounds on the Number of Maximal Subgroups of Finite Groups: Applications

by Adolfo Ballester-Bolinches, Ramón Esteban-Romero and Paz Jiménez-Seral

Mathematics 2022, 10(7), 1153; https://doi.org/10.3390/math10071153 - 2 Apr 2022

Cited by 3 | Viewed by 3105

Abstract

The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given probability. In this paper, we obtain new bounds [...] Read more.

The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given probability. In this paper, we obtain new bounds for the number of maximal subgroups of a given index in a finite group and we pin-point the universal constants that appear in some results in the literature related to the number of maximal subgroups of a finite group with a given index. This allows us to compare properly our bounds with some of the known bounds. Full article

(This article belongs to the Special Issue Group Theory and Related Topics)

20 pages, 6612 KiB

Open AccessEditor’s ChoiceArticle

Proving Feasibility of a Docking Mission: A Contractor Programming Approach

by Auguste Bourgois, Simon Rohou, Luc Jaulin and Andreas Rauh

Mathematics 2022, 10(7), 1130; https://doi.org/10.3390/math10071130 - 1 Apr 2022

Cited by 1 | Viewed by 2434

Abstract

Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this [...] Read more.

Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this approach is limited as it is based on parameter space discretization and therefore cannot be considered a proof of feasibility. To overcome this limitation, set-membership methods based on interval analysis, guaranteed integration, and contractor programming have proven their efficiency. Guaranteed integration algorithms can predict the possible trajectories of a system initialized in a given set in the form of tubes of trajectories. The contractor programming consists in removing the trajectories violating predefined constraints from a system’s tube of possible trajectories. Our contribution consists in merging both approaches to allow for the usage of differential constraints in a contractor programming framework. We illustrate our method through examples related to robotics. We also released an open-source implementation of our algorithm in a unified library for tubes, allowing one to combine it with other constraints and increase the number of possible applications. Full article

(This article belongs to the Special Issue Set-Based Methods for Differential Equations and Applications)

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12 pages, 3181 KiB

Open AccessEditor’s ChoiceArticle

A Dynamic Mechanistic Model of Perceptual Binding

by Pavel Kraikivski

Mathematics 2022, 10(7), 1135; https://doi.org/10.3390/math10071135 - 1 Apr 2022

Cited by 5 | Viewed by 2946

Abstract

The brain’s ability to create a unified conscious representation of an object by integrating information from multiple perception pathways is called perceptual binding. Binding is crucial for normal cognitive function. Some perceptual binding errors and disorders have been linked to certain neurological conditions, [...] Read more.

The brain’s ability to create a unified conscious representation of an object by integrating information from multiple perception pathways is called perceptual binding. Binding is crucial for normal cognitive function. Some perceptual binding errors and disorders have been linked to certain neurological conditions, brain lesions, and conditions that give rise to illusory conjunctions. However, the mechanism of perceptual binding remains elusive. Here, I present a computational model of binding using two sets of coupled oscillatory processes that are assumed to occur in response to two different percepts. I use the model to study the dynamic behavior of coupled processes to characterize how these processes can modulate each other and reach a temporal synchrony. I identify different oscillatory dynamic regimes that depend on coupling mechanisms and parameter values. The model can also discriminate different combinations of initial inputs that are set by initial states of coupled processes. Decoding brain signals that are formed through perceptual binding is a challenging task, but my modeling results demonstrate how crosstalk between two systems of processes can possibly modulate their outputs. Therefore, my mechanistic model can help one gain a better understanding of how crosstalk between perception pathways can affect the dynamic behavior of the systems that involve perceptual binding. Full article

(This article belongs to the Special Issue Dynamics and Bifurcations in Mathematical Neuroscience: Analysis, Modeling and Applications)

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18 pages, 4281 KiB

Open AccessEditor’s ChoiceArticle

Adaptive Fuzzy Neural Network Harmonic Control with a Super-Twisting Sliding Mode Approach

by Qi Pan, Xiangguo Li and Juntao Fei

Mathematics 2022, 10(7), 1063; https://doi.org/10.3390/math10071063 - 25 Mar 2022

Cited by 6 | Viewed by 2447

Abstract

This paper designed an adaptive super-twisting sliding mode control (STSMC) scheme based on an output feedback fuzzy neural network (OFFNN) for an active power filter (APF), aiming at tracking compensation current quickly and precisely, and solving the harmonic current problem in the electrical [...] Read more.

This paper designed an adaptive super-twisting sliding mode control (STSMC) scheme based on an output feedback fuzzy neural network (OFFNN) for an active power filter (APF), aiming at tracking compensation current quickly and precisely, and solving the harmonic current problem in the electrical grid. With the use of OFFNN approximator, the proposed controller has the characteristic of full regulation and high approximation accuracy, where the parameters of OFFNN can be adjusted to the optimal values adaptively, thereby increasing the versatility of the control method. Moreover, due to an added signal feedback loop, the controller can obtain more information to track the state variable faster and more correctly. Simulations studies are given to demonstrate the performance of the proposed controller in the harmonic suppression, and verify its better steady-state and dynamic performance. Full article

(This article belongs to the Special Issue Advances in Intelligent Control)

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16 pages, 490 KiB

Open AccessEditor’s ChoiceArticle

A Modified Grey Wolf Optimization Algorithm for an Intrusion Detection System

by Abdullah Alzaqebah, Ibrahim Aljarah, Omar Al-Kadi and Robertas Damaševičius

Mathematics 2022, 10(6), 999; https://doi.org/10.3390/math10060999 - 21 Mar 2022

Cited by 103 | Viewed by 6917

Abstract

Cyber-attacks and unauthorized application usage have increased due to the extensive use of Internet services and applications over computer networks, posing a threat to the service’s availability and consumers’ privacy. A network Intrusion Detection System (IDS) aims to detect aberrant traffic behavior that [...] Read more.

Cyber-attacks and unauthorized application usage have increased due to the extensive use of Internet services and applications over computer networks, posing a threat to the service’s availability and consumers’ privacy. A network Intrusion Detection System (IDS) aims to detect aberrant traffic behavior that firewalls cannot detect. In IDSs, dimension reduction using the feature selection strategy has been shown to be more efficient. By reducing the data dimension and eliminating irrelevant and noisy data, several bio-inspired algorithms have been employed to improve the performance of an IDS. This paper discusses a modified bio-inspired algorithm, which is the Grey Wolf Optimization algorithm (GWO), that enhances the efficacy of the IDS in detecting both normal and anomalous traffic in the network. The main improvements cover the smart initialization phase that combines the filter and wrapper approaches to ensure that the informative features will be included in early iterations. In addition, we adopted a high-speed classification method, the Extreme Learning Machine (ELM), and used the modified GWO to tune the ELM’s parameters. The proposed technique was tested against various meta-heuristic algorithms using the UNSWNB-15 dataset. Because the generic attack is the most common attack type in the dataset, the primary goal of this paper was to detect generic attacks in network traffic. The proposed model outperformed other methods in minimizing the crossover error rate and false positive rate to less than 30%. Furthermore, it obtained the best results with 81%, 78%, and 84% for the accuracy, F1-score, and G-mean measures, respectively. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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21 pages, 4688 KiB

Open AccessEditor’s ChoiceArticle

Evaluation of sEMG Signal Features and Segmentation Parameters for Limb Movement Prediction Using a Feedforward Neural Network

by David Leserri, Nils Grimmelsmann, Malte Mechtenberg, Hanno Gerd Meyer and Axel Schneider

Mathematics 2022, 10(6), 932; https://doi.org/10.3390/math10060932 - 15 Mar 2022

Cited by 4 | Viewed by 3148

Abstract

Limb movement prediction based on surface electromyography (sEMG) for the control of wearable robots, such as active orthoses and exoskeletons, is a promising approach since it provides an intuitive control interface for the user. Further, sEMG signals contain early information about the onset [...] Read more.

Limb movement prediction based on surface electromyography (sEMG) for the control of wearable robots, such as active orthoses and exoskeletons, is a promising approach since it provides an intuitive control interface for the user. Further, sEMG signals contain early information about the onset and course of limb movements for feedback control. Recent studies have proposed machine learning-based modeling approaches for limb movement prediction using sEMG signals, which do not necessarily require domain knowledge of the underlying physiological system and its parameters. However, there is limited information on which features of the measured sEMG signals provide the best prediction accuracy of machine learning models trained with these data. In this work, the accuracy of elbow joint movement prediction based on sEMG data using a simple feedforward neural network after training with different single- and multi-feature sets and data segmentation parameters was compared. It was shown that certain combinations of time-domain and frequency-domain features, as well as segmentation parameters of sEMG data, improve the prediction accuracy of the neural network as compared to the use of a standard feature set from the literature. Full article

(This article belongs to the Section E: Applied Mathematics)

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21 pages, 365 KiB

Open AccessEditor’s ChoiceArticle

Riemann–Hilbert Problems and Soliton Solutions of Type (

λ^{*}

,

- λ^{*}

) Reduced Nonlocal Integrable mKdV Hierarchies

by Wen-Xiu Ma

Mathematics 2022, 10(6), 870; https://doi.org/10.3390/math10060870 - 9 Mar 2022

Cited by 55 | Viewed by 3323

Abstract

Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter

λ

with its complex conjugate

λ^{*}

, and the [...] Read more.

Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter

λ

with its complex conjugate

λ^{*}

, and the other one is nonlocal, which replaces the spectral parameter

λ

with its negative complex conjugate

- λ^{*}

. Riemann–Hilbert problems and thus inverse scattering transforms are formulated from the reduced matrix spectral problems. In view of the specific distribution of eigenvalues and adjoint eigenvalues, soliton solutions are constructed from the reflectionless Riemann–Hilbert problems. Full article

13 pages, 1455 KiB

Open AccessEditor’s ChoiceArticle

A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers

by Vahid Andalib and Jyotirmoy Sarkar

Mathematics 2022, 10(6), 852; https://doi.org/10.3390/math10060852 - 8 Mar 2022

Cited by 20 | Viewed by 3441

Abstract

Assuming exponential lifetime and repair time distributions, we study the limiting availability

A_{\infty}

as well as the per unit time-limiting profit

ω

of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either [...] Read more.

Assuming exponential lifetime and repair time distributions, we study the limiting availability

A_{\infty}

as well as the per unit time-limiting profit

ω

of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either by an in-house repairer within an exponential patience time T or by an external expert who works faster but charges more. When there are two repair facilities, we allow the regular repairer to begin repair or to continue repair beyond T if the expert is busy. Two models arise accordingly as the expert repairs one or all failed units during each visit. We show that (1) adding a second spare to a one-unit system already backed by a spare raises

A_{\infty}

as well as

ω

; (2) thereafter, adding a second repair facility improves both criteria further. Finally, we determine whether the expert must repair one or all failed units to maximize these criteria and fulfill the maintenance management objectives better than previously studied models. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation II)

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14 pages, 268 KiB

Open AccessEditor’s ChoiceArticle

Intermediate-Task Transfer Learning with BERT for Sarcasm Detection

by Edoardo Savini and Cornelia Caragea

Mathematics 2022, 10(5), 844; https://doi.org/10.3390/math10050844 - 7 Mar 2022

Cited by 58 | Viewed by 6950

Abstract

Sarcasm detection plays an important role in natural language processing as it can impact the performance of many applications, including sentiment analysis, opinion mining, and stance detection. Despite substantial progress on sarcasm detection, the research results are scattered across datasets and studies. In [...] Read more.

Sarcasm detection plays an important role in natural language processing as it can impact the performance of many applications, including sentiment analysis, opinion mining, and stance detection. Despite substantial progress on sarcasm detection, the research results are scattered across datasets and studies. In this paper, we survey the current state-of-the-art and present strong baselines for sarcasm detection based on BERT pre-trained language models. We further improve our BERT models by fine-tuning them on related intermediate tasks before fine-tuning them on our target task. Specifically, relying on the correlation between sarcasm and (implied negative) sentiment and emotions, we explore a transfer learning framework that uses sentiment classification and emotion detection as individual intermediate tasks to infuse knowledge into the target task of sarcasm detection. Experimental results on three datasets that have different characteristics show that the BERT-based models outperform many previous models. Full article

(This article belongs to the Special Issue Natural Language Processing (NLP) and Machine Learning (ML)—Theory and Applications)

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18 pages, 786 KiB

Open AccessEditor’s ChoiceArticle

How Many Fractional Derivatives Are There?

by Duarte Valério, Manuel D. Ortigueira and António M. Lopes

Mathematics 2022, 10(5), 737; https://doi.org/10.3390/math10050737 - 25 Feb 2022

Cited by 41 | Viewed by 4918

Abstract

In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also [...] Read more.

In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also some myths of Fractional Calculus and false fractional derivatives. The results are expected to contribute to limit the appearance of derivatives that differ from existing ones just because they are defined on distinct domains, and to prevent the ambiguous use of the concept of fractional derivative. Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications (DSTA) — In Memory of Prof. Dr. José A. Tenreiro Machado)

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17 pages, 3315 KiB

Open AccessEditor’s ChoiceArticle

PFVAE: A Planar Flow-Based Variational Auto-Encoder Prediction Model for Time Series Data

by Xue-Bo Jin, Wen-Tao Gong, Jian-Lei Kong, Yu-Ting Bai and Ting-Li Su

Mathematics 2022, 10(4), 610; https://doi.org/10.3390/math10040610 - 16 Feb 2022

Cited by 113 | Viewed by 13120

Abstract

Prediction based on time series has a wide range of applications. Due to the complex nonlinear and random distribution of time series data, the performance of learning prediction models can be reduced by the modeling bias or overfitting. This paper proposes a novel [...] Read more.

Prediction based on time series has a wide range of applications. Due to the complex nonlinear and random distribution of time series data, the performance of learning prediction models can be reduced by the modeling bias or overfitting. This paper proposes a novel planar flow-based variational auto-encoder prediction model (PFVAE), which uses the long- and short-term memory network (LSTM) as the auto-encoder and designs the variational auto-encoder (VAE) as a time series data predictor to overcome the noise effects. In addition, the internal structure of VAE is transformed using planar flow, which enables it to learn and fit the nonlinearity of time series data and improve the dynamic adaptability of the network. The prediction experiments verify that the proposed model is superior to other models regarding prediction accuracy and proves it is effective for predicting time series data. Full article

(This article belongs to the Special Issue Mathematical Method and Application of Machine Learning)

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20 pages, 4930 KiB

Open AccessEditor’s ChoiceArticle

Forecasting of Electrical Energy Consumption in Slovakia

by Michal Pavlicko, Mária Vojteková and Oľga Blažeková

Mathematics 2022, 10(4), 577; https://doi.org/10.3390/math10040577 - 12 Feb 2022

Cited by 26 | Viewed by 4444

Abstract

Prediction of electricity energy consumption plays a crucial role in the electric power industry. Accurate forecasting is essential for electricity supply policies. A characteristic feature of electrical energy is the need to ensure a constant balance between consumption and electricity production, whereas electricity [...] Read more.

Prediction of electricity energy consumption plays a crucial role in the electric power industry. Accurate forecasting is essential for electricity supply policies. A characteristic feature of electrical energy is the need to ensure a constant balance between consumption and electricity production, whereas electricity cannot be stored in significant quantities, nor is it easy to transport. Electricity consumption generally has a stochastic behavior that makes it hard to predict. The main goal of this study is to propose the forecasting models to predict the maximum hourly electricity consumption per day that is more accurate than the official load prediction of the Slovak Distribution Company. Different models are proposed and compared. The first model group is based on the transverse set of Grey models and Nonlinear Grey Bernoulli models and the second approach is based on a multi-layer feed-forward back-propagation network. Moreover, a new potential hybrid model combining these different approaches is used to forecast the maximum hourly electricity consumption per day. Various performance metrics are adopted to evaluate the performance and effectiveness of models. All the proposed models achieved more accurate predictions than the official load prediction, while the hybrid model offered the best results according to performance metrics and supported the legitimacy of this research. Full article

(This article belongs to the Special Issue Statistical Data Modeling and Machine Learning with Applications II)

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18 pages, 1418 KiB

Open AccessEditor’s ChoiceArticle

Data-Driven Maintenance Priority and Resilience Evaluation of Performance Loss in a Main Coolant System

by Hongyan Dui, Zhe Xu, Liwei Chen, Liudong Xing and Bin Liu

Mathematics 2022, 10(4), 563; https://doi.org/10.3390/math10040563 - 11 Feb 2022

Cited by 28 | Viewed by 2500

Abstract

The main coolant system (MCS) plays a vital role in the stability and reliability of a nuclear power plant. However, human errors and natural disasters may cause some reactor coolant system components to fail, resulting in severe consequences such as nuclear leakage. Therefore, [...] Read more.

The main coolant system (MCS) plays a vital role in the stability and reliability of a nuclear power plant. However, human errors and natural disasters may cause some reactor coolant system components to fail, resulting in severe consequences such as nuclear leakage. Therefore, it is crucial to perform a resilience analysis of the MCS, to effectively reduce and prevent losses. In this paper, a resilience importance measure (RIM) for performance loss is proposed to evaluate the performance of the MCS. Specifically, a loss importance measure (LIM) is first proposed to indicate the component maintenance priority of the MCS under different failure conditions. Based on the LIM, RIMs for single component failure and multiple component failures were developed to measure the recovery efficiency of the system performance. Finally, a case study was conducted to demonstrate the proposed resilience measure for system reliability. Results provide a valuable reference for increasing the system security of the MCS and choosing the appropriate total maintenance cost. Full article

(This article belongs to the Special Issue Data-Driven Methods and Artificial Intelligence in Reliability and Maintenance)

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19 pages, 869 KiB

Open AccessEditor’s ChoiceArticle

Variational Bayesian Inference in High-Dimensional Linear Mixed Models

by Jieyi Yi and Niansheng Tang

Mathematics 2022, 10(3), 463; https://doi.org/10.3390/math10030463 - 31 Jan 2022

Cited by 8 | Viewed by 4298

Abstract

In high-dimensional regression models, the Bayesian lasso with the Gaussian spike and slab priors is widely adopted to select variables and estimate unknown parameters. However, it involves large matrix computations in a standard Gibbs sampler. To solve this issue, the Skinny Gibbs sampler [...] Read more.

In high-dimensional regression models, the Bayesian lasso with the Gaussian spike and slab priors is widely adopted to select variables and estimate unknown parameters. However, it involves large matrix computations in a standard Gibbs sampler. To solve this issue, the Skinny Gibbs sampler is employed to draw observations required for Bayesian variable selection. However, when the sample size is much smaller than the number of variables, the computation is rather time-consuming. As an alternative to the Skinny Gibbs sampler, we develop a variational Bayesian approach to simultaneously select variables and estimate parameters in high-dimensional linear mixed models under the Gaussian spike and slab priors of population-specific fixed-effects regression coefficients, which are reformulated as a mixture of a normal distribution and an exponential distribution. The coordinate ascent algorithm, which can be implemented efficiently, is proposed to optimize the evidence lower bound. The Bayes factor, which can be computed with the path sampling technique, is presented to compare two competing models in the variational Bayesian framework. Simulation studies are conducted to assess the performance of the proposed variational Bayesian method. An empirical example is analyzed by the proposed methodologies. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

12 pages, 274 KiB

Open AccessEditor’s ChoiceArticle

Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

by Simona Decu and Stefan Haesen

Mathematics 2022, 10(3), 330; https://doi.org/10.3390/math10030330 - 21 Jan 2022

Cited by 7 | Viewed by 3127

Abstract

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality [...] Read more.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

15 pages, 731 KiB

Open AccessEditor’s ChoiceArticle

Comparison of the Selected Methods Used for Solving the Ordinary Differential Equations and Their Systems

by Edyta Hetmaniok and Mariusz Pleszczyński

Mathematics 2022, 10(3), 306; https://doi.org/10.3390/math10030306 - 19 Jan 2022

Cited by 11 | Viewed by 3138

Abstract

Ordinary differential equations (ODEs), and the systems of such equations, are used for describing many essential physical phenomena. Therefore, the ability to efficiently solve such tasks is important and desired. The goal of this paper is to compare three methods devoted to solving [...] Read more.

Ordinary differential equations (ODEs), and the systems of such equations, are used for describing many essential physical phenomena. Therefore, the ability to efficiently solve such tasks is important and desired. The goal of this paper is to compare three methods devoted to solving ODEs and their systems, with respect to the quality of obtained solutions, as well as the speed and reliability of working. These approaches are the classical and often applied Runge–Kutta method of order 4 (RK4), the method developed on the ground of the Taylor series, the differential transformation method (DTM), and the routine available in the Mathematica software (Mat). Full article

(This article belongs to the Special Issue Applications of Symbolic and Soft Computations in Applied Sciences)

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19 pages, 329 KiB

Open AccessEditor’s ChoiceArticle

Applications of Solvable Lie Algebras to a Class of Third Order Equations

by María S. Bruzón, Rafael de la Rosa, María L. Gandarias and Rita Tracinà

Mathematics 2022, 10(2), 254; https://doi.org/10.3390/math10020254 - 14 Jan 2022

Cited by 3 | Viewed by 2243

Abstract

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for [...] Read more.

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs. Full article

(This article belongs to the Special Issue Symmetry Methods and Applications for Nonlinear Partial Differential Equations)

28 pages, 3306 KiB

Open AccessEditor’s ChoiceArticle

Parameter Identification of Photovoltaic Models by Hybrid Adaptive JAYA Algorithm

by Xiaobing Yu, Xuejing Wu and Wenguan Luo

Mathematics 2022, 10(2), 183; https://doi.org/10.3390/math10020183 - 7 Jan 2022

Cited by 21 | Viewed by 2290

Abstract

As one of the most promising forms of renewable energy, solar energy is increasingly deployed. The simulation and control of photovoltaic (PV) systems requires identification of their parameters. A Hybrid Adaptive algorithm based on JAYA and Differential Evolution (HAJAYADE) is developed to identify [...] Read more.

As one of the most promising forms of renewable energy, solar energy is increasingly deployed. The simulation and control of photovoltaic (PV) systems requires identification of their parameters. A Hybrid Adaptive algorithm based on JAYA and Differential Evolution (HAJAYADE) is developed to identify these parameters accurately and reliably. The HAJAYADE algorithm consists of adaptive JAYA, adaptive DE, and the chaotic perturbation method. Two adaptive coefficients are introduced in adaptive JAYA to balance the local and global search. In adaptive DE, the Rank/Best/1 mutation operator is put forward to boost the exploration and maintain the exploitation. The chaotic perturbation method is applied to reinforce the local search further. The HAJAYADE algorithm is employed to address the parameter identification of PV systems through five test cases, and the eight latest meta-heuristic algorithms are its opponents. The mean RMSE values of the HAJAYADE algorithm from five test cases are 9.8602 × 10⁻⁴, 9.8294 × 10⁻⁴, 2.4251 × 10⁻³, 1.7298 × 10⁻³, and 1.6601 × 10⁻². Consequently, HAJAYADE is proven to be an efficient and reliable algorithm and could be an alternative algorithm to identify the parameters of PV systems. Full article

(This article belongs to the Special Issue Applied Computing and Artificial Intelligence)

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17 pages, 910 KiB

Open AccessEditor’s ChoiceArticle

Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures

by Sergey A. Lurie, Dmitrii B. Volkov-Bogorodskii and Petr A. Belov

Mathematics 2022, 10(1), 90; https://doi.org/10.3390/math10010090 - 27 Dec 2021

Cited by 2 | Viewed by 2758

Abstract

A mathematical statement for the coupled stationary thermoelasticity is given on the basis of a variational approach and the contact boundary problem is formulated to consider inhomogeneous materials. The structure of general representation of the solution from the set of the auxiliary potentials [...] Read more.

A mathematical statement for the coupled stationary thermoelasticity is given on the basis of a variational approach and the contact boundary problem is formulated to consider inhomogeneous materials. The structure of general representation of the solution from the set of the auxiliary potentials is established. The potentials are analyzed depending on the parameters of the model, taking into account the restrictions associated with additional requirements for the positive definiteness of the potential energy density for the coupled problem in the one-dimensional case. The novelty of this work lies in the fact that it attempts to take into account the effects of higher order coupling between the gradients of the temperature fields and the gradients of the deformation fields. From a mathematical point of view, this leads to a change in the roots of the characteristic equation and affects the structure of the solution. Contact boundary value problems are formulated for modeling inhomogeneous materials and a solution for a layered structure is constructed. The analysis of the influence of the model parameters on the structure of the solution is given. The features of the distribution of mechanical and thermal fields in the region of phase contact with a change in the parameters, which are characteristic only for gradient theories of coupled thermoelasticity and stationary thermal conductivity, are discussed. It is shown, for example, that taking into account the additional parameter of connectivity of gradient fields of deformations and temperatures predicts the appearance of rapidly changing temperature fields and significant localization of heat fluxes in the vicinity of phase contact in inhomogeneous materials. Full article

(This article belongs to the Section E6: Functional Interpolation)

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20 pages, 351 KiB

Open AccessEditor’s ChoiceArticle

Negative Order KdV Equation with No Solitary Traveling Waves

by Miguel Rodriguez, Jing Li and Zhijun Qiao

Mathematics 2022, 10(1), 48; https://doi.org/10.3390/math10010048 - 24 Dec 2021

Cited by 9 | Viewed by 2641

Abstract

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve [...] Read more.

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions. Full article

(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)

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21 pages, 3512 KiB

Open AccessEditor’s ChoiceArticle

Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

by Humam Kareem Jalghaf, Endre Kovács, János Majár, Ádám Nagy and Ali Habeeb Askar

Mathematics 2021, 9(24), 3308; https://doi.org/10.3390/math9243308 - 19 Dec 2021

Cited by 17 | Viewed by 4264

Abstract

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the [...] Read more.

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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20 pages, 2112 KiB

Open AccessEditor’s ChoiceArticle

Deep Learning Models for Predicting Monthly TAIEX to Support Making Decisions in Index Futures Trading

by Duy-An Ha, Chia-Hung Liao, Kai-Shien Tan and Shyan-Ming Yuan

Mathematics 2021, 9(24), 3268; https://doi.org/10.3390/math9243268 - 16 Dec 2021

Cited by 2 | Viewed by 5663

Abstract

Futures markets offer investors many attractive advantages, including high leverage, high liquidity, fair, and fast returns. Highly leveraged positions and big contract sizes, on the other hand, expose investors to the risk of massive losses from even minor market changes. Among the numerous [...] Read more.

Futures markets offer investors many attractive advantages, including high leverage, high liquidity, fair, and fast returns. Highly leveraged positions and big contract sizes, on the other hand, expose investors to the risk of massive losses from even minor market changes. Among the numerous stock market forecasting tools, deep learning has recently emerged as a favorite tool in the research community. This study presents an approach for applying deep learning models to predict the monthly average of the Taiwan Capitalization Weighted Stock Index (TAIEX) to support decision-making in trading Mini-TAIEX futures (MTX). We inspected many global financial and economic factors to find the most valuable predictor variables for the TAIEX, and we examined three different deep learning architectures for building prediction models. A simulation on trading MTX was then performed with a simple trading strategy and two different stop-loss strategies to show the effectiveness of the models. We found that the Temporal Convolutional Network (TCN) performed better than other models, including the two baselines, i.e., linear regression and extreme gradient boosting. Moreover, stop-loss strategies are necessary, and a simple one could be sufficient to reduce a severe loss effectively. Full article

(This article belongs to the Special Issue Mathematics and Financial Economics)

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9 pages, 457 KiB

Open AccessEditor’s ChoiceArticle

Generalized Confidence Intervals for Zero-Inflated Pareto Distribution

by Xiao Wang and Xinmin Li

Mathematics 2021, 9(24), 3272; https://doi.org/10.3390/math9243272 - 16 Dec 2021

Cited by 7 | Viewed by 3227

Abstract

This paper considers interval estimations for the mean of Pareto distribution with excess zeros. Three approaches for interval estimation are proposed based on fiducial generalized pivotal quantities (FGPQs), respectively. Simulation studies are performed to assess the performance of the proposed methods, along with [...] Read more.

This paper considers interval estimations for the mean of Pareto distribution with excess zeros. Three approaches for interval estimation are proposed based on fiducial generalized pivotal quantities (FGPQs), respectively. Simulation studies are performed to assess the performance of the proposed methods, along with three measurements to determine comparisons with competing approaches. The advantages and disadvantages of each method are provided. The methods are illustrated using a real phone call dataset. Full article

(This article belongs to the Special Issue Advances in Computational Statistics and Applications)

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25 pages, 1653 KiB

Open AccessEditor’s ChoiceArticle

COSMONET: An R Package for Survival Analysis Using Screening-Network Methods

by Antonella Iuliano, Annalisa Occhipinti, Claudia Angelini, Italia De Feis and Pietro Liò

Mathematics 2021, 9(24), 3262; https://doi.org/10.3390/math9243262 - 15 Dec 2021

Cited by 5 | Viewed by 4909

Abstract

Identifying relevant genomic features that can act as prognostic markers for building predictive survival models is one of the central themes in medical research, affecting the future of personalized medicine and omics technologies. However, the high dimension of genome-wide omic data, the strong [...] Read more.

Identifying relevant genomic features that can act as prognostic markers for building predictive survival models is one of the central themes in medical research, affecting the future of personalized medicine and omics technologies. However, the high dimension of genome-wide omic data, the strong correlation among the features, and the low sample size significantly increase the complexity of cancer survival analysis, demanding the development of specific statistical methods and software. Here, we present a novel R package, COSMONET (COx Survival Methods based On NETworks), that provides a complete workflow from the pre-processing of omics data to the selection of gene signatures and prediction of survival outcomes. In particular, COSMONET implements (i) three different screening approaches to reduce the initial dimension of the data from a high-dimensional space p to a moderate scale d, (ii) a network-penalized Cox regression algorithm to identify the gene signature, (iii) several approaches to determine an optimal cut-off on the prognostic index (

P I

) to separate high- and low-risk patients, and (iv) a prediction step for patients’ risk class based on the evaluation of

P I s

. Moreover, COSMONET provides functions for data pre-processing, visualization, survival prediction, and gene enrichment analysis. We illustrate COSMONET through a step-by-step R vignette using two cancer datasets. Full article

(This article belongs to the Special Issue Computational Approaches for Data Inspection in Biomedicine)

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42 pages, 1092 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

The Representation Theory of Neural Networks

by Marco Armenta and Pierre-Marc Jodoin

Mathematics 2021, 9(24), 3216; https://doi.org/10.3390/math9243216 - 13 Dec 2021

Cited by 13 | Viewed by 6771

Abstract

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver [...] Read more.

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work. Full article

(This article belongs to the Special Issue Mathematics, Statistics and Applied Computational Methods)

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19 pages, 8868 KiB

Open AccessEditor’s ChoiceArticle

Global Value Chains of COVID-19 Materials: A Weighted Directed Network Analysis

by Georgios Angelidis, Charalambos Bratsas, Georgios Makris, Evangelos Ioannidis, Nikos C. Varsakelis and Ioannis E. Antoniou

Mathematics 2021, 9(24), 3202; https://doi.org/10.3390/math9243202 - 11 Dec 2021

Cited by 11 | Viewed by 3300

Abstract

The COVID-19 pandemic caused a boom in demand for personal protective equipment, or so-called “COVID-19 goods”, around the world. We investigate three key sectoral global value chain networks, namely, “chemicals”, “rubber and plastics”, and “textiles”, involved in the production of these goods. First, [...] Read more.

The COVID-19 pandemic caused a boom in demand for personal protective equipment, or so-called “COVID-19 goods”, around the world. We investigate three key sectoral global value chain networks, namely, “chemicals”, “rubber and plastics”, and “textiles”, involved in the production of these goods. First, we identify the countries that export a higher value added share than import, resulting in a “value added surplus”. Then, we assess their value added flow diversification using entropy. Finally, we analyze their egonets in order to identify their key affiliates. The relevant networks were constructed from the World Input-Output Database. The empirical results reveal that the USA had the highest surplus in “chemicals”, Japan in “rubber and plastics”, and China in “textiles”. Concerning value added flows, the USA was highly diversified in “chemicals”, Germany in “rubber and plastics”, and Italy in “textiles”. From the analysis of egonets, we found that the USA was the key supplier in all sectoral networks under consideration. Our work provides meaningful conclusions about trade outperformance due to the fact of surplus, trade flow robustness due to the fact of diversification, and trade partnerships due to the egonets analysis. Full article

(This article belongs to the Special Issue Complex Network Modeling: Theory and Applications)

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16 pages, 1578 KiB

Open AccessEditor’s ChoiceArticle

Optimization of Sliding Mode Control to Save Energy in a SCARA Robot

by Luis Arturo Soriano, José de Jesús Rubio, Eduardo Orozco, Daniel Andres Cordova, Genaro Ochoa, Ricardo Balcazar, David Ricardo Cruz, Jesus Alberto Meda-Campaña, Alejandro Zacarias and Guadalupe Juliana Gutierrez

Mathematics 2021, 9(24), 3160; https://doi.org/10.3390/math9243160 - 8 Dec 2021

Cited by 44 | Viewed by 6765

Abstract

Sliding mode control is a robust technique that is used to overcome difficulties such as parameter variations, unmodeled dynamics, external disturbances, and payload changes in the position-tracking problem regarding robots. However, the selection of the gains in the controller could produce bigger forces [...] Read more.

Sliding mode control is a robust technique that is used to overcome difficulties such as parameter variations, unmodeled dynamics, external disturbances, and payload changes in the position-tracking problem regarding robots. However, the selection of the gains in the controller could produce bigger forces than are required to move the robots, which requires spending a large amount of energy. In the literature, several approaches were used to manage these features, but some proposals are complex and require tuning the gains. In this work, a sliding mode controller was designed and optimized in order to save energy in the position-tracking problem of a two-degree-of-freedom SCARA robot. The sliding mode controller gains were optimized usinga Bat algorithm to save energy by minimizing the forces. Finally, two controllers were designed and implemented in the simulation, and as a result, adequate controller gains were found that saved energy by minimizing the forces. Full article

(This article belongs to the Special Issue Numerical Simulation and Control in Energy Systems)

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12 pages, 259 KiB

Open AccessEditor’s ChoiceArticle

Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

by Slavko Simić and Vesna Todorčević

Mathematics 2021, 9(23), 3104; https://doi.org/10.3390/math9233104 - 1 Dec 2021

Cited by 6 | Viewed by 2108

Abstract

In this article, we give sharp two-sided bounds for the generalized Jensen functional

J_{n} (f, g, h;

p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean [...] Read more.

In this article, we give sharp two-sided bounds for the generalized Jensen functional

J_{n} (f, g, h;

p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean

A_{n} (h;

p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

9 pages, 374 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations

by Nikolay A. Kudryashov

Mathematics 2021, 9(23), 3024; https://doi.org/10.3390/math9233024 - 25 Nov 2021

Cited by 68 | Viewed by 3254

Abstract

Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method [...] Read more.

Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method for finding solutions of equation is given in the general case. Solitary waves of equation are obtained as implicit function taking into account the transformation of variables. Full article

(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)

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9 pages, 255 KiB

Open AccessEditor’s ChoiceArticle

Semi-Hyers–Ulam–Rassias Stability of the Convection Partial Differential Equation via Laplace Transform

by Daniela Marian

Mathematics 2021, 9(22), 2980; https://doi.org/10.3390/math9222980 - 22 Nov 2021

Cited by 17 | Viewed by 2637

Abstract

In this paper, we study the semi-Hyers–Ulam–Rassias stability and the generalized semi-Hyers–Ulam–Rassias stability of some partial differential equations using Laplace transform. One of them is the convection partial differential equation. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)

13 pages, 302 KiB

Open AccessEditor’s ChoiceArticle

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

by Helena Myšková and Ján Plavka

Mathematics 2021, 9(22), 2951; https://doi.org/10.3390/math9222951 - 18 Nov 2021

Cited by 2 | Viewed by 1977

Abstract

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there [...] Read more.

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability. Full article

(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)

18 pages, 2213 KiB

Open AccessEditor’s ChoiceArticle

A Bayesian-Deep Learning Model for Estimating COVID-19 Evolution in Spain

by Stefano Cabras

Mathematics 2021, 9(22), 2921; https://doi.org/10.3390/math9222921 - 16 Nov 2021

Cited by 17 | Viewed by 3099

Abstract

This work proposes a semi-parametric approach to estimate the evolution of COVID-19 (SARS-CoV-2) in Spain. Considering the sequences of 14-day cumulative incidence of all Spanish regions, it combines modern Deep Learning (DL) techniques for analyzing sequences with the usual Bayesian Poisson-Gamma model for [...] Read more.

This work proposes a semi-parametric approach to estimate the evolution of COVID-19 (SARS-CoV-2) in Spain. Considering the sequences of 14-day cumulative incidence of all Spanish regions, it combines modern Deep Learning (DL) techniques for analyzing sequences with the usual Bayesian Poisson-Gamma model for counts. The DL model provides a suitable description of the observed time series of counts, but it cannot give a reliable uncertainty quantification. The role of expert elicitation of the expected number of counts and its reliability is DL predictions’ role in the proposed modelling approach. Finally, the posterior predictive distribution of counts is obtained in a standard Bayesian analysis using the well known Poisson-Gamma model. The model allows to predict the future evolution of the sequences on all regions or estimates the consequences of eventual scenarios. Full article

(This article belongs to the Special Issue Advances in Computational Statistics and Applications)

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